Inputs, outputs and states in the representation of time series

Inputs, outputs and states in the representation of time series

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Praagman, C. (1987). Inputs, outputs and states in the representation of time series. (Memorandum COSOR; Vol. 8732). Technische Universiteit Eindhoven.

Document status and date: Published: 01/01/1987

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EINDHOVEN UNIVERSITY OF TECHNOLOGY Department of Mathematics and Computing Science

Memorandum COSOR 87-32 Inputs, outputs and states in the

representation of time series by

C. Praagman

Eindhoven University of Technology

Department of Mathematics and Computing Science P.O. Box 513

5600 MB Eindhoven The Netherlands

Eindhoven, October 1987 The Netherlands

INTRODUCTION

Recently the development of numerically reliable algorithms for polynomial matrices has dmwn the attention within the field of system theory. See for instance Van DOOREN [4], BEELEN [1], BEELEN-VELTKAMP [2], BEELEN-van den HURK-PRAAGMAN [3]. In this paper I shall apply these recently achieved results to obtain various representations for families of time series, specified a priori by AR-equations.

AJ.:t

algorithms

are

numerically reliable, but some

are

very time consuming.

§ 1. Polynomial operators in the backward and/or forward shift

The starting point for my investigations is a family of time series, defined on Z, with values in Rq, defined as the solutions of an AR-equation:

Rdlf(t +d)+ ... +R~(t)=O.

Denoting the backward shift: (R q

f

~ (R q)z by a:

(O'}f)(t)

=

If(t

+

I) ,

and defining the polynomial matrix R E Rgxl/ [s] by

R (s)

=

RdSd + ... + R 0 , this equation can be rewritten as:

R(O)lf =0 .

In the sequel I shall denote the solution space of this equation by B (R ): B(R) = {If E (Rq)z : R(o)w = O} .

Since

a

is an invertible opemtor it might be more natural to consider R E RQX8 [s, 8-1]. In the sequel I shall use both points of view. Two matrices R and R'

are

equivalent if there exist

U, U' with entries in R [s ,s-l] such that UR

=

R' and R

=

U'R' (Wll.,LEMS [6]). So espe-cially it is clear that premultiplying R by an orthogonal matrix, or multiplying a row by sl,

for an I E Z does not change B (R). That deleting a number of zero rows in R does not change B (R) is already obvious from more direct observation

Finally I give some definitions concerning polynomial matrices. Let R E R'XIl [8 ,8-1] be given by

2

-Then Rd is called the leading coefficient matrix of R and Ric: the trailing coefficient matrix. If R = (r': ... : rq ). ri e Rgx1[s ,S-I], then Rei, the leading column coefficient matrix is defined as:

Rei = (r'l: ... : rli), r/ the leading coefficient matrix of ri

Analogously the trailing column coefficient matrix Ret' the leading and trailing row coefficient matrices Rrl and Ret

are

defined.

R is called row (column) reduced if Rrl(Rcl ) has full row (column) rank. If moreover

Rrt(Rct } has full row (column) rank R is called bilaterally row (column) reduced.

§ 2. Review of some algorithms

In this paper I shall describe some algorithms concerning B (R). All algorithms have a name and

are

followed by two sets of parameters seperated by a semicolon. The first set of parameters

are

the inputs of the algorithm, and the second the output. In this section I have collected a number of algorithms well-known and/or derived elsewhere which I shall use.

ROWCOMPRESS (A; B ,p)

For a constant matrix A e Rm'hl an orthogonal matrix B e R mxm and an integer p is

detennined such

thaI

BA

=

~'].

with A' e RP"" of full row rank.

COLCOMPRESS (A; B .p )

Analogous but now AB = (0: A ').

ROWDELETE (A; B ,p )

For an A e Rm'hl [s, s-l] aBe RP'hI [s, s-l] is detennined by deleting all zero rows in

A.

3

-Deletes zero columns in A .

MINRPOLBAS (P;N)

For a JX>lynomial P E Rm><Ji [s] a polynomial matrix N E R1I.><P [s] is determined, with

p

=

n

-rank P , such that

PN =0.

N is column reduced,

N (s) has full column rank for all seq; .

The columns of N form a basis of ker P in R (s )n. and among all polynomial bases of ker P , its total degree is minimal. The description of this algorithm and a discussion of its reliability can be found in BEELEN-VELTKAMP [2].

MINLPOLBAS (P;N)

=

MINRPOLBAS (PT,NT).

COLEMBED (P; U).

For a JX>lynomial matrix P e Rm><Ji [s], having full column rank for all seq; , a unimo-dular U is determined (det U e R *) such that U

=

(P : Q ).

This algorithm can be found in BEELEN [1].

ROWEMBED (P; U)

=

COLEMBED (pT; UT ).

MULTIPLY (P ,p ,1;P,).

Let U = diag(lp;

s'

Im-p), then P'

=

UP. ROWRED (P;P').

For aPe Rnl><Ji [s] of full row rank a unimodular U e RnlXl1l. [s] is determined such that P'

=

UP is row reduced.

4

-This algorithm is described in detail in BEELEN-van den HURK-PRAAGMAN [3].

COLSPLIT (P ,p ; P', P ").

For a P e R mXII [s ,s-I], and an integer p ,P' e RmXP [s, S-I] and

P"

e

Rmx(n-p)[s ,S-I]

are

defined by P = (P':P").

ROWSPLIT (P,p;P',P'')

=

COLSPLIT (pT,p;p,T,p"T).

§ 3. AR-rnodels

So

I

start with B(R) = {~ e (Rq)z : R«(J)~ =

OJ.

If one wants to minimize the total degree (= sum of the degrees) of the equation determining B (R), one should look for a bila-terally row reduced R' such that B(R')

=

B(R). It is well-known that such

an

R' always exists (WILLEMS [6]). The following algorithm determines one:

algorithm 1. BILROWRED (R ,R '); determination of a bilaterally row reduced R' equivalent toRe Rgxq[s].

BEGIN

MINLPOLBAS (R;N); }

ROWEMBED (N; U); determines

an

equivalent R" of full row rank ROWDELETE (UR;R";g');

WHILE rank R"o ¢ g' }

ROWCOMPRESS (R"o;V,p) Changes R" such that RUt = R"o has full row rank MULTIPLY (VR",p ,-l;R");

ROWRED (R";R'); END;

Note that rowreducing R" does not affect the rank of R" 0: R'o

=

R'(O)

=

U(O)R"(O)

=

U,fl"o. and since U is unimodular det _{U o}

=

det U ¢ O. Although

a bilaterally row reduced representation is very desirable for constructing inputs/outputs and minimal states, a serious drawback is the rather slow algorithm ROWRED; therefore I look for alternatives in the following sections.

5

-§ 4. Input-output representations.

Let B (R) be as above. A triple (P. Q • T) e R'XP [8] X R8><1n [8] X GL_q R is an input-output representation for B (R) if

rank P

=

rankR

=

P

(Q:-P)T =R

P+Q is proper (p+ = (pT p)-lpT)

Soif7W

=

~].thenB(R)= I"~]

1l'J. =Ql!).

The following algorithm detennines a triple (P, Q , T):

algorithm 2. lOR (R; P ,Q, T); detennination of an Input-Output Representation (P , Q ,T) for

B(R). BEGIN MINRPOLBAS (R; N); ROWCOMPRESS (N_{cl ;} T, k); COLSPLIT (RT-1, k; Q ,-P); END;

Note that since N is column reduced. TN also is. and hence if TN

= [;:].

N 2 N,' is (strictly) proper. From (Q:-P)T-1TN =QN1-PNz=O it follows that P+Q

=

NzN'il is (strictly) proper, too.

§ 5. States

A state space description for B (R) is a family of timeseries B,

c

(R" ~ x (Rq)z such that:

1£ e B(R) if and only ifthereexists.I e (R"~ suchthat(A.1£)e B,.

If(A.~)e B,,(A',w')e B, and.I(t)=.I'(t) then (A,w) 1\ (x'.~')e B, t

faCt)

if

t

<

to

Here (g "to Q)(t) =

l

_b(t) if t ~ to .

6

-EXAMPLE. Let R (8) = Rdsd + ... + r Q. Define n = dq and

l

W(t) ]

,,(t) = : .

(t - d)

Starting with a B (R) it is natural to look for externally induced states: .I

=

f

~). Especially I shall look for f's defined by a matrix X E RItXll [s. S-I]. In the above example

X(s)=

Further it is desirable to have an X which reflects exactly when two time series 1£ and w' can be glued together: X is called faithfull if (Xl£) (0)

=

0 if and only if 0 " 1£ E B (R ).

Q

The following algorithm determines a faithfull X E Rdq [8-1].

algorithm 3. FSD (R; X); determination of a Faithfull X determining a Statespace Description for B(R).

Xes)

=

It is straightforward to verify that X is a faithfull state inducing map.

Now consider for any X the subspace X(B(R» of (RIt)E. From wn..LEMS [6] it follows that there exists an R' E

R

rXII

[8] such that X (B (R»

=

B (R

J.

Since moreover .I is a state variable there is an R' of first degree: R '(8) = R' IS

+

R' 0- TIle following algorithm determines R' 1 and R' 0:

algorithm 4. ARS (R. X; R '); determination of an AR -relation for the state X (B (R».

BEGIN

LetskX = Y E RItXll[8], and degree Y = l; ex := 3d

+

I;

MINLPOUlAS

[~~R];

(L:

-U) ]

R':s-«L; END;

-7-Comparing this algorithm with ROWRED as described in BEELEN-van den HURK-PRAAGMAN [3] will show that it works, but is not very fast

I shall not go in detail here.

A more promising approach is to look for an AR relation for Bs .

It is known (WILLEMS [7]) that there exist E ,F E Rtxn and G E Rtxq such that

Bs

=

{(A, w) E (Rq)1l : EoX

+

FX

+

Gw

=

O}

For X given by FSD (R; X) E ,F and G are easily found:

ox=

f~ ~lx+ [~olw,andRd}£+(o",I)X=O

l

I 0 Rd- 1

describes

B,

completely. so

E

=

[~}F

=

[~]

and G

=

[:~]

with

f

=

(d

+

I)

q.

§ 6.lnput-state-output representations

If in the final description of Bs in the preceding section Rd would be of full row rank,

then Rd, '

=

(Qd:-Pd) with Pd invertible. So taking Tw

= [:]

this leads to an "onlinary" input-state-output description of Bs by matrices A ,B, C .D:

B,

=

Ii;<.

w) : CJ;l

=

A.l

+

Bll.)'

=

eX

+DK.

~]

=

Tw} .

The final algorithm I shall give calculates a representation of this type with D = 0 for B (R ).

algorithm S. ISO (R ; A, B , C ,T); determination of an Input-Slate-OUtput representation

8 -BEGIN

REPEAT

ROWDELETE (R; R ,p);

ROWCOMPRESS _(Rd; U. r); making Rd of full row rank

MULTIPLY (UR,r,-l;R); UNTIL rank Rd

=

P ;

COLCOMPRESS (R_{d ;} T ,p ) COLSPLIT (Rd ,-p ; Qd' -P d) R := Pd-1R; COLSPLIT (R .p; Q, P);

detemination of input and output.

A>:L

_l~

~O]'B:=[~O]

I P d-l Qd-l

C := (0' .. 0 J) END;

If one looks at the equation at the end of the preceding section, it is clear that replacing

w by "

~]

leads to

o

0

1 0

0 1 0

but fonn the fonn of Rd

=

(0: -I) one sees that Rdl!!. + (0 ... I)x.

=

0 is

equal

to

1'.

=

(0 ... I)x..

As a last observation note that the state space is not trim: there are many states through which no state trajectory passes.

§

7. Final remarks

In this paper I considered timeseries defined on Z. Note that all algorithms can be easily adapted to the case where B(R)

c

(R<i)N. except for algorithm 5 which completely fails.

9

-More detailed proofs and considerations concerning these algorithms will be presented in PRAAGMAN [5], which is under preparation~

References

[1] BEELEN. Th.GJ. New algorithms for computing the Kronecker structure of a pencil with applications to

systems and control theory. Thesis, Eindhoven 1987. [2] BEELEN, Th.G.J. & Numerical computation of a coprime factorization

G.W. Veltkamp of a transfer function matrix. Systems and Control Letters 9 (1987).

[3]

BEELEN, Th.GJ. & A new method for computing a column reduced G.J.H. van den HURK & polynomial matrix. Preprint, Eindhoven 1987 C. PRAAGMAN

[4] DOOREN, P. van The computation of Kronecker's canonical fonn

of a singular pencil. Lin. Alg. Appl. 27 (1974) 103-140. [5] PRAAGMAN, C. Algorithms to associate inputs, outputs and states to

AR -models. In preparation.

[6] WILLEMS, J.C. From time series to linear system. part 1: Finite dimensional linear time invariant systems. Automatica 22 (1986) 561-580.